Modeling a cylindrical object to resemble an image reference
1- Model the main object and the shape to use for Boolean modifier.

2- Select the main object, in Modifiers panel, add a Boolean modifier, assign the object you want to extract under (Object) in Boolean modifier.

3- Choose Difference as the Method, then Apply the modifier.
4- Delete the extraction object or move it to another layer.

The disadvantage of this method is that it'll create Ngons, yet since this a radial symmetrical object, you can model 1/8th of the object first, use the Boolean modifier and apply it, fix the Ngons by connecting vertices using J or using knife tool K, then copy rotate it 45 degrees, in order to get the whole object.